Kinetic Equations and Fluctuations in Nonideal Gases and Plasmas

  • Werner EbelingEmail author
  • Vladimir E. Fortov
  • Vladimir Filinov
Part of the Springer Series in Plasma Science and Technology book series (SSPST)


This brief survey of kinetic and transport theory starts with the stochastic approach, which is mathematically simpler than kinetic equations of Boltzmann and Lenard–Balescu type. We continue with Lorentz approximations and then discuss the derivation of quantum Boltzmann equations using the Bogoliubov method. The next topic is fluctuation–dissipation theorems, including a discussion of recent controversies in this field, and several applications to plasmas. For more detailed introductions to stochastic and transport problems, the reader is referred to other books (Balescu, Equilibrium and Nonequilibrium Statistical Mechanics, 1975; Klimontovich, 1982, Kinetic theory of nonideal gases and nonideal plasmas, 1986; Ebeling et al., 1984, 2002, 2005; Dittrich et al., Quantum Transport and Dissipation, 1998; Bonitz, Progress in nonequilibrium Green’s Functions, 2000, Quantum Kinetic Theory, B.G. Teubner, Stuttgart 1998, 2016; Kremp et al., Quantum Statistics of Nonideal Plasmas, 2005).


  1. S.V. Adamjan, I.M. Tkachenko, J.L. Munoz-Cobo, G. Verdu, Martin, Dynamic and static correlations in model Coulomb systems. Phys. Rev. E 48, 2067–2072 (1993)ADSCrossRefGoogle Scholar
  2. N. Ashcroft. N.D. Mermin, Solid State Physics (Holt, Rinehardt and Winston, Philadelphia 1976, Mir Moscow 1979)Google Scholar
  3. R. Balescu, Equilibrium And Nonequilibrium Statistical Mechanics (Wiley, New York, 1975)zbMATHGoogle Scholar
  4. K. Binder, G. Cicotti (eds.), The Monte Carlo and Molecular Dynamics of Condensed Matter Systems (SIF, Bologna, 1996)Google Scholar
  5. N.N. Bogoliubov, N.N Bogoliubov Jr, Introduction to Quantum Statistical Mechanics (Gordon and Breach, New York, 1991)Google Scholar
  6. N.N. Bogoliubov, Selected Works, vol (II, Quantum and Classical Statistical Mechanics (Gordon and Breach, New York, 1991)Google Scholar
  7. N.N. Bogoliubov, Collected Papers. vol. 1–12. (Fizmatlit, Moscow 2005–2009)Google Scholar
  8. M. Bonitz (ed.), Progress In Nonequilibrium Green’s Functions (World Scientific, Singapore, 2000)zbMATHGoogle Scholar
  9. M. Bonitz, Quantum kinetic theory (Springer, Berlin, 2016)CrossRefzbMATHGoogle Scholar
  10. H.B. Böttger, V.V. Bryksin, Hopping Conduction in Solids (Akademie-Verlag, Berlin, 1985)Google Scholar
  11. L. Brizhik, A.P. Chetverikov, W. Ebeling, G. Röpke, M.G. Velarde, Electron pairing and Coulomb repulsion in one-dimensional anharmonic lattices. Phys. Rev. B 85, 245105 (2012)ADSCrossRefGoogle Scholar
  12. H.B. Callen, T.A. Welton, Irreversibility and generalized noise. Phys. Rev. 83, 34–40 (1951)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. R. Car, M. Parrinello, Unified Approach for Molecular Dynamics and Density-Functional Theory. Phys. Rev. Lett. 55, 2471–247 (1985)ADSCrossRefGoogle Scholar
  14. P. Chetverikov, W. Ebeling, G. Röpke, M.G. Velarde, Anharmonic excitations, time correlations and electric conductivity. Contrib. Plasma Phys. 47, 465–478 (2007)ADSCrossRefGoogle Scholar
  15. A. Chetverikov, W. Ebeling, M.G. Velarde, Local electron distributions and diffusion in anharmonic lattices. Eur. Phys. J. B 70, 217–227 (2009)ADSCrossRefGoogle Scholar
  16. A.P. Chetverikov, W. Ebeling, G. Röpke, Hopping transport and stochastic dynamics of electrons in plasma layers. Contr. Plasma Phys. 51, 814–829 (2011)ADSCrossRefGoogle Scholar
  17. A.P. Chetverikov, W. Ebeling, M.G. Velarde, in Without bounds a scientific canvas of nonlinearity and complex dynamics, ed. by R.G. Rubio, et al., Toward a theory of degenerated solectrons in doped lattices (Springer, Berlin, 2013)Google Scholar
  18. A.P. Chetverikov, W. Ebeling, M.G. Velarde, On the temperature dependence of fast electron transport in lattices. Eur. Phys. J. B 88, 202 (2014)ADSCrossRefGoogle Scholar
  19. A.P. Chetverikov, W. Ebeling, M.G. Velarde, Controlling fast electron transfer at the nano-scale by solitonic excitations along crystallographic axes. Eur. Phys. J. B 85, 291 (2015)ADSCrossRefGoogle Scholar
  20. A.S. Davydov, Solitons In Molecular Systems (Reidel, Dordrecht, 1991)CrossRefzbMATHGoogle Scholar
  21. C. Dorso, S. Duarte, J. Randrup, Excited electron dynamics modeling of warm matter, Phys. Lett. B 188 287 (1987); 215, 611 (1988)Google Scholar
  22. T. Dittrich, P. Hänggi, G. Ingold, B. Kramer, G. Schön, W. Zwerger, Quantum Transport And Dissipation (Wiley-VCH, Weinheim, 1998)zbMATHGoogle Scholar
  23. W. Ebeling, Bound state effects in quantum transport theory. Ann. Physik (Leipzig) 33, 350–358 (1976)ADSCrossRefGoogle Scholar
  24. W. Ebeling, G. Röpke, Conductance theory of nonideal plasmas. Ann. Physik (Leipzig) 36, 429–432 (1979)ADSCrossRefGoogle Scholar
  25. W. Ebeling V.E. Fortov et al. (eds.), Transport Properties of Dense Plasmas (Birkhäuser, Basel, Boston 1984)Google Scholar
  26. W. Ebeling, Förster, Thermodynamic, kinetics and phase transitions in dense plasmas, In: Elementary processes in dense plasmas (S. Ichimaru, S. Ogata, eds.), Addison-Wesley Reading 1995Google Scholar
  27. W. Ebeling, I. Sokolov, Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems (World Scientific, Singapore, 2005)CrossRefzbMATHGoogle Scholar
  28. W. Ebeling, A. Filinov, M. Bonitz, V. Filinov, T. Pohl, The method of effective potentials in the quantum-statistical theory of plasmas. J. Phys. A Math. Gen. 39, 4309–4317 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. W. Ebeling, E. Gudowska, I. Sokolov, On stochastic dynamics in physics: History and terminology. Acta Phys. Pol. 39, 1003–1018 (2008)ADSMathSciNetzbMATHGoogle Scholar
  30. R. Feistel, W. Ebeling, Physics Of Self-organization And Evolution (Wiley, Weinheim, 2011)CrossRefzbMATHGoogle Scholar
  31. V.L. Ginzburg, L.D. Landau, Zh Eksp, Teor. Fiz. 20, 1064 (1950)Google Scholar
  32. V.L. Ginzburg, L.P. Pitaevskii, Quantum Nyquist formula, applicability ranges of the Callen-Welton formula. Sov. Phys. Uspekhi 30, 168–171 (1987)ADSMathSciNetCrossRefGoogle Scholar
  33. L.P. Gorkov, Microscopic derivation of the Ginzburg-Landau equations for superconductivity, Sov. Phys. JETP E, 1364 (1959), JETP 36, 1918 (1959)Google Scholar
  34. J.-P. Hansen, I.R. McDonald, E.L. Pollock, Phys. Rev. A 11, 1025 (1975)ADSCrossRefGoogle Scholar
  35. J.-P. Hansen, I.K. McDonald, Phys. Rev. A 23, 2041 (1981)ADSCrossRefGoogle Scholar
  36. N.M. Hugenholtz, Derivation of the Boltzmann equation for a Fermi gas. J. Stat. Phys. 32, 231–254 (1983)ADSMathSciNetCrossRefGoogle Scholar
  37. S. Ichimaru, Statistical Plasma Physics (Addison-Wesley, Redwood City, 1992)Google Scholar
  38. J.C. Inkson, Many-body Theory Of Solids (Plenum Press, New York, 1984)CrossRefGoogle Scholar
  39. Y.L. Klimontovich, Statistical theory of nonequilibrium processes in plasmas, Izdat MGU, Moskva 1964 (Engl. transl. Pergamon, Oxford, 1967)Google Scholar
  40. Y.L. Klimontovich, Kinetic theory of nonideal gases and nonideal plasmas(in Russ.), (Nauka, Moskva 1975, Engl. transl. Pergamon Press, Oxford 1982)Google Scholar
  41. Y.L. Klimontovich, Statistical Physics(in Russ.) (Nauka, Moscow 1982, Engl. transl. Harwood 1986)Google Scholar
  42. Y.L. Klimontovich, Statistical Theory of Open Systems (Kluwer, Amsterdam, 1987)Google Scholar
  43. Y.L. Klimontovich, Statistical Theory of Open Systems (in Russian). vol. I, II, III (Janus, Moscow 1995, 1999, 2001)Google Scholar
  44. V.P. Yu, Klimontovich, Silin, The spectra of systems of interacting particles and collective losses during passage of charged particles through matter. Soviet Physics Uspekhi 3, 84–114 (1960)Google Scholar
  45. W. Yu, Klimontovich, Ebeling, Quantum kinetic equations of nonideal gases and nonideal plasma. Zh. eksp. teor. Fiz. 63, 905–917 (1972)Google Scholar
  46. W.D. Kraeft, D. Kremp, W. Ebeling, G. Röpke, Quantum Statistics of Charged Particle Systems, (Akademie & Pergamon Press, Berlin & New York 1986)Google Scholar
  47. D. Kremp, M. Schlanges, W.D. Kraeft, Quantum statistics of nonideal plasmas (Springer, Berlin, 2005)zbMATHGoogle Scholar
  48. R. Kubo, Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. J. Phys. Soc. Jpn. 12, 570–586 (1957)ADSMathSciNetCrossRefGoogle Scholar
  49. J.D. Landau, E.M. Lifshits, Statistical Physics (part I), Nauka Moskva 1976, German transl. Berlin 1979, Engl. transl. Statistical Physics, Part 1. vol. 5, 3rd ed. Butterworth Heinemann 1980Google Scholar
  50. L.D. Landau, Collected papers (Oxford: Pergamon Press, 1965) (Nauka, Moscow, 1969)Google Scholar
  51. L.D. Landau, E.M. Lifshits. Quantum mechanics: Nonrelativistic theory, Moscow Nauka 1976, Engl. translation Pergamon Press, Oxford 1977, German transl. Akademie Verlag Berlin 1977Google Scholar
  52. A.S. Larkin, V.S. Filinov, V.E. Fortov, Path integral representation of the Wigner function. Contr. Plasma Phys. 56(3–4), 197–213 (1916)Google Scholar
  53. E.M. Lifshits, P. Pitaevskii, Physical Kinetics, Course of Theoretical Physics, vol. 10 (Pergamon. New York, 1981)Google Scholar
  54. B. Militzer, E.L. Pollock, Phys. Rev. E 61, 3470 (2000)ADSCrossRefGoogle Scholar
  55. E. Montroll, J. Ward, Quantum statistics of interacting particles. Phys. Fluids 1, 55 (1958)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. W.J. Nellis, S.T. Weir, A.C. Mitchell, Phys. Rev. B 59, 3434 (1999)ADSCrossRefGoogle Scholar
  57. H. Nyquist, Thermal agitation of electric charge in conductors. Phys. Rev. 32, 110–113 (1929)ADSCrossRefGoogle Scholar
  58. J. Ortner, F. Schautz, W. Ebeling, Quasiclassical molecular dynamics simulations of the electron gas: Dynamic properties. Phys. Rev. E 56, 4665 (1997)ADSCrossRefGoogle Scholar
  59. J. Ortner, I. Valuev, W. Ebeling, Sermiclassical dynamics and time correlations in two-component plasmas. Contr. Plasma Phys. 39, 311–321 (1999)ADSCrossRefGoogle Scholar
  60. J. Ortner, I. Valuev and W. Ebeling, Microfield Distribution in TCP, Contr. Plasma Phys. 40, 555–685 (2000). W. Pauli, Über das H-theorem vom Anwachsen der Entropie vom Standpunkt der neuen Quantenmechanik. In: Probleme der Modernen Physik (P. Debye, ed.), pp. 30–45, Leipzig Hirzel 1928Google Scholar
  61. D. Pines, The Many-body Problem. A Lecture Note (Benjamin, New York 1961), (Russ. transl., Moskva 1963)Google Scholar
  62. D. Pines, P. Nozieres, The theory of quantum liquids(Benjamin, New York, Amsterdam 1966) (Russ. Transl. Mir, Moskva, 1967)Google Scholar
  63. R. Redmer, G. Röpke, Progress in the theory of dense strongly coupled plasmas. Contr. Plasma Phys. 50, 970–985 (2010)ADSCrossRefGoogle Scholar
  64. H. Reinholz, G. Röpke, S. Rosmej, R. Redmer, Conductivity of warm dense matter including electron-electron collisions. Phys. Rev. E 91, 043105 (2015)ADSCrossRefGoogle Scholar
  65. G. Röpke, W. Ebeling, W.D. Kraeft, Quantum-statistical conductance theory of nonideal plasmas by use of the force-force correlation function method. Physica A 101, 243–254 (1980)ADSCrossRefGoogle Scholar
  66. S.P. Sadykova, W. Ebeling, I.M. Tkachenko, Eur. Phys. J. D 61, 117–130 (2011)ADSCrossRefGoogle Scholar
  67. V.P. Silin, A.A. Rukhadse, Electromagnetic Properties of plasmas and plasma-like matter(in Russ.). (Atomisdat, Moskva 1961)Google Scholar
  68. L. Spitzer, Physics of fully ionized plasmas (Wiley, New York, 1961) (Russ. transl, Mir Moskva, 1965)Google Scholar
  69. O.G. Sitenko, Fluctuations And Nonlinear Wave Interactions In Plasmas (Pergamon Press, Oxford, 1982)zbMATHGoogle Scholar
  70. R.L. Stratonovich, Nonlinear Nonequilibrium Thermodynamics I, Linear and Nonlinear Fluctuation-Dissipation Theorems (Springer, Heidelberg, 1994)zbMATHGoogle Scholar
  71. W. Thirring, Lehrbuch der Mathematischen Physik. 4 Quantenmechanik großer Systeme Springer 1980Google Scholar
  72. R.C. Tolman, The Principles Of Statistical Mechanics (University Press, Oxford, 1938)Google Scholar
  73. V.M. Zamalin, G.E. Norman, V.S. Filinov, The Monte Carlo Method in Statistical Thermodynamics (Nauka, Moscow, 1977). (in Russian)Google Scholar
  74. J. Von Neumann, Mathematische Grundlagen der Quantenmechanik (Springer, Berlin, 1932)zbMATHGoogle Scholar
  75. V.M. Zamalin, G.E. Norman, V.S. Filinov, Monte Carlo method in statistical mechanics (in Russ.), (Nauka, Moscow, 1977)Google Scholar
  76. D.N Zubarev, V. Morozov, G. Röpke, Statistical Mechanics of Nonequilibrium Processes (Wiley VCH, Weinheim 1996, 1997)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Werner Ebeling
    • 1
    Email author
  • Vladimir E. Fortov
    • 2
  • Vladimir Filinov
    • 3
  1. 1.Institut für PhysikHumboldt Universität BerlinBerlinGermany
  2. 2.Russian Academy of SciencesMoscowRussia
  3. 3.Joint Institute for High TemperaturesRussian Academy of SciencesMoscowRussia

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